Why is AM=MB=MC in this right triangle?

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I understand that AM=BC because M is the centerpoint, but I do not understand MC.

2 Answers
Jul 20, 2018

Please refer to the Explanation.

Explanation:

MM is the midpoint of ABAB and /_ACB=90^@ACB=90.

Therefore, CC lies on the (semi)circle described on AB,AB, taking ABAB

as diameter.

MM is the midpoint of the diameter ABAB.

:. M is the centre of the circle.

Thus, C lies on the circle having centre at M and a diameter AB.

Clearly, MA=MB=MC"(=the radius of the circle)".

Jul 20, 2018

Please see below.

Explanation:

Your Question :
I understand that AM=color(red)(BC) because M is

the centerpoint, but I do not understand MC.

I think it is AM=MB and not color(red)(BC)

We have two Theorem :

color(green)((1) "An angle inscribed in a semicircle is a right angle"

color(green)((2)"The circle whose diameter is the hypotenuse of the "

color(green)("right triangle , passes through three vertices of the triangle."

enter image source here

So, A ,B, and C are the points on the circle and mangleC=90^circ

:.color(violet)( " Hypotenuse AB of triangle = Diameter AB of circle"

Now , M " is the midpoint of AB"

So, "M is the center of the circle. "

Let color(blue)(r) be the radius of circle.

:.color(blue)(AM= MB=r

Here, bar(MC) is the line segment joining center M and vertex C

:. MC " is the radius of circle."

:.color(blue)( MC =r)

Hence ,

color(blue)(AM=MB=MC=r